Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laboratory

A General Theory of Semi-Unification

Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. We introduce a general case of semi-unification where substitutions are allowed on non-regular terms, and we prove the equivalence of this general case to a well-known undecidable data base dependency problem, thus establishing the undecidability of general semi-unification. We present a unified way of looking at the various problems of semi-unification. We give some properties that are common to all the cases of semi-unification. We also the principality property and the solution set for those problems. We prove that semi-unification on general terms has the principality property. Finally, we present a recursive inseparability result between semi-unification on regular terms and semi-unification on general terms.

A Direct Algorithm for the Type Interference in the Rank 2 Fragment of the Second--Order λ-Calculus

We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k ≥ 3 of this stratification. While it was already known that typability is decidable at rank ≤ 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show how to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-unification. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-unification.

Type Reconstruction in the Presence of Polymorphic Recursion and the Recursive Types

We establish the equivalence of type reconstruction with polymorphic recursion and recursive types is equivalent to regular semi-unification which proves the undecidability of the corresponding type reconstruction problem. We also establish the equivalence of type reconstruction with polymorphic recursion and positive recursive types to a special case of regular semi-unification which we call positive regular semi-unification. The decidability of positive regular semi-unification is an open problem.

The Undecidability of Mitchell's Subtyping Relationship

Mitchell defined and axiomatized a subtyping relationship (also known as containment, coercibility, or subsumption) over the types of System F (with "→" and "∀"). This subtyping relationship is quite simple and does not involve bounded quantification. Tiuryn and Urzyczyn quite recently proved this subtyping relationship to be undecidable. This paper supplies a new undecidability proof for this subtyping relationship. First, a new syntax-directed axiomatization of the subtyping relationship is defined. Then, this axiomatization is used to prove a reduction from the undecidable problem of semi-unification to subtyping. The undecidability of subtyping implies the undecidability of type checking for System F extended with Mitchell's subtyping, also known as "F plus eta".

Beta-Reduction As Unification

We define a unification problem ^UP with the property that, given a pure lambda-term M, we can derive an instance Gamma(M) of ^UP from M such that Gamma(M) has a solution if and only if M is beta-strongly normalizable. There is a type discipline for pure lambda-terms that characterizes beta-strong normalization; this is the system of intersection types (without a "top" type that can be assigned to every lambda-term). In this report, we use a lean version LAMBDA of the usual system of intersection types. Hence, ^UP is also an appropriate unification problem to characterize typability of lambda-terms in LAMBDA. It also follows that ^UP is an undecidable problem, which can in turn be related to semi-unification and second-order unification (both known to be undecidable).